Dedekind Cut

A set partition of the Rational Numbers into two nonempty subsets $S_1$ and $S_2$ such that all members of $S_1$ are less than those of $S_2$ and such that $S_1$ has no greatest member. Real Numbers can be defined using either Dedekind cuts or Cauchy Sequences.

See also Cantor-Dedekind Axiom, Cauchy Sequence

References

Courant, R. and Robbins, H. ``Alternative Methods of Defining Irrational Numbers. Dedekind Cuts.'' §2.2.6 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 71-72, 1996.